We can represent a function by its graph, that is, by the points:
{(x,y) | y=f(x) } If we reverse the points, we can ask the question what function is represented by
{(y,x) | y=f(x) } The answer is, the inverse function of f . The graph is the same as that of f(x), but rotated about the 45 degree line, y=x.
The inverse function will pass the vertical line test, if and only if the function passes the horizontal line test:
Definition: Horizontal Line Test. A function passes the horizontal line test if and only if no horizontal line intersects its graph more than once.A function which passes both horizontal and vertical line tests is said to be one-to-one , that is f(a)=f(b) if and only if a=b. We can formally solve the equation y=f(x) by stating that x=f-1(y). Note: This does not mean x=1/f(y)!f-1(x) is a function with the property that f(f-(x))=x=f-1(f(x))
We can find f-1(x) as a function by the following three step rule:
This differs slightly from the text, and interchanges steps 2 and 3...
- Write y=f(x)
- Interchange x and y, x=f(y)
- Solve for y in terms of x to get y=f-1(x)
Theorem. If f is one-to-one and continuous, then f-1(x) is also.
Theorem. If f is one-to-one and differentiable, with g(x)= f-1(x) , then the derivative of g(x) is given by g'(a)=1/f'(g(a)).
The latter theorem is a direct result of differentiating x=g(f(x)) by means of the chain rule...